A wizard with a long beard and bright, flaming eyes places a table in front of you. On the table are two circular rings resembling frisbees. The wizard motions to the table. "We call them Galadriel rings after the great elf of Middle Earth, and we use the rings to test humans with whom we come in contact. As you see, one ring is red, the other gold. Both rings are the same size. The red ring is stuck to the table. The gold ring rotates around the red ring, touching it without slipping. When the gold ring has completed a turn around the red one, how many turns has it made around its own axis?"
To determine how many turns the gold ring has made around its own axis when it completes one full rotation around the red ring, we need to examine the specific properties and movement of the two rings.
Since the gold ring rotates around the red ring without slipping, it means that their points of contact stay constantly in contact with each other. This implies that the gold ring moves along the circumference of the red ring.
Now, let's consider the relationship between the circumferences of the two rings. Since both rings are the same size, their circumferences are equal.
When the gold ring completes one full rotation around the red ring, it travels a distance equal to the circumference of the red ring. At the same time, the gold ring also completes one full rotation around its own axis.
Therefore, when the gold ring completes one turn around the red ring, it also completes one turn around its own axis.
To determine the number of turns the gold ring has made around its own axis after completing a turn around the red ring, we need to consider the relative motion between the two rings.
First, let's establish a reference frame. We can imagine ourselves looking down at the rings from above the table. In this frame, we consider the rotation of the gold ring around its own axis to be clockwise.
Now, as the wizard mentioned, the gold ring is rotating around the red ring without slipping. This means that the point on the gold ring that touches the red ring will travel along the circumference of the red ring during its rotation.
Since the gold ring completes a full turn around the red ring, the point touching the red ring will trace the circumference of the red ring once. This implies that the gold ring also rotates around its own axis by the same distance covered on the red ring's circumference.
Let's assume the red ring has a circumference of 10 units for simplicity. As the gold ring completes its turn around the red ring, it will cover the same distance, which is also 10 units, in terms of the gold ring's own circumference. Therefore, the gold ring makes one complete turn around its own axis.
So, to answer the wizard's question, when the gold ring completes a turn around the red ring, it has also made one turn around its own axis.